openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /FiniteWordJointPadding.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartPadding | |
| namespace OAI | |
| /-! # Padding preserves the joint law with every unused suffix bit -/ | |
| namespace ContingencyTables.FiniteObservableTransport | |
| open CommonBasesFPRAS FairBits | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| theorem SameLaw.append_joint {Z : Type*} {n q : ℕ} | |
| {f : (Fin n → Bool) → Z} {g : (Fin q → Bool) → Z} | |
| (h : SameLaw f g) (t : ℕ) : | |
| SameLaw (fun bits : Fin (n+t) → Bool => | |
| (f (splitWordEquiv n t bits).1,(splitWordEquiv n t bits).2)) | |
| (fun bits : Fin (q+t) → Bool => | |
| (g (splitWordEquiv q t bits).1,(splitWordEquiv q t bits).2)) := by | |
| have hp := h.prod (show SameLaw (fun bits : Fin t → Bool => bits) (fun bits : Fin t → Bool => bits) | |
| from fun _ => rfl) | |
| intro phi | |
| have hl := uniformAverage_equiv (splitWordEquiv n t) | |
| (fun p : (Fin n → Bool)×(Fin t → Bool) => phi (f p.1,p.2)) | |
| have hr := uniformAverage_equiv (splitWordEquiv q t) | |
| (fun p : (Fin q → Bool)×(Fin t → Bool) => phi (g p.1,p.2)) | |
| exact hl.trans ((hp phi).trans hr.symm) | |
| theorem restart_append_joint_sameLaw | |
| (base : List CompanionRestartProgram.Stage×List ℕ) (n q : ℕ) | |
| (hn : (∑ i : Fin base.1.length,CompanionRestartProgram.widthAt base i.val)=n) | |
| (g : (Fin q → Bool) → List ℕ) | |
| (h : SameLaw (fun bits : Fin n → Bool => | |
| CompanionRestartProgram.output (base,List.ofFn bits)) g) (t : ℕ) : | |
| SameLaw (fun bits : Fin (n+t) → Bool => | |
| (CompanionRestartProgram.output (base,List.ofFn bits),(splitWordEquiv n t bits).2)) | |
| (fun bits : Fin (q+t) → Bool => | |
| (g (splitWordEquiv q t bits).1,(splitWordEquiv q t bits).2)) := by | |
| have hp := h.append_joint t | |
| have he (bits : Fin (n+t) → Bool) : | |
| CompanionRestartProgram.output (base,List.ofFn bits)= | |
| CompanionRestartProgram.output (base,List.ofFn (splitWordEquiv n t bits).1) := by | |
| have hs := (CompanionRestartProgram.output_take_sum base (List.ofFn bits)).symm | |
| rw [hn,DensePrograms.ofFn_split_take] at hs | |
| exact hs | |
| intro phi | |
| simp_rw [he] | |
| exact hp phi | |
| /-- Flattening the first two independent blocks preserves their joint | |
| law with the untouched third block. -/ | |
| theorem nested_word_joint {Z : Type*} (a b t : ℕ) | |
| (f : (Fin a → Bool) → (Fin b → Bool) → Z) : | |
| SameLaw (fun bits : Fin (a+(b+t)) → Bool => | |
| (f (splitWordEquiv a (b+t) bits).1 | |
| (splitWordEquiv b t (splitWordEquiv a (b+t) bits).2).1, | |
| (splitWordEquiv b t (splitWordEquiv a (b+t) bits).2).2)) | |
| (fun bits : (Fin (a+b) → Bool)×(Fin t → Bool) => | |
| (f (splitWordEquiv a b bits.1).1 (splitWordEquiv a b bits.1).2,bits.2)) := by | |
| intro phi | |
| have houter := uniformAverage_equiv (splitWordEquiv a (b+t)) | |
| (fun p : (Fin a → Bool)×(Fin (b+t) → Bool) => | |
| phi (f p.1 (splitWordEquiv b t p.2).1,(splitWordEquiv b t p.2).2)) | |
| have hinner (u : Fin a → Bool) := uniformAverage_equiv (splitWordEquiv b t) | |
| (fun p : (Fin b → Bool)×(Fin t → Bool) => phi (f u p.1,p.2)) | |
| have hfirst := uniformAverage_equiv (splitWordEquiv a b) | |
| (fun p : (Fin a → Bool)×(Fin b → Bool) => | |
| uniformAverage (fun tail : Fin t → Bool => phi (f p.1 p.2,tail))) | |
| rw [uniformAverage_prod] at hfirst | |
| rw [uniformAverage_prod] | |
| refine houter.trans ?_ | |
| rw [uniformAverage_prod] | |
| calc | |
| _ = uniformAverage (fun u : Fin a → Bool => | |
| uniformAverage (fun v : Fin b → Bool => | |
| uniformAverage (fun tail : Fin t → Bool => phi (f u v,tail)))) := by | |
| apply congrArg uniformAverage | |
| funext u | |
| exact (hinner u).trans (uniformAverage_prod _) | |
| _ = _ := hfirst.symm | |
| end | |
| end ContingencyTables.FiniteObservableTransport | |
| end OAI | |